I am in DC and have walked around again. The density feels right for a city, much like Tokyo, London and Paris (all notable for a lack of overly tall buildings). In DC, the buildings are not too tall and canyon like, and there are few vacant lots in the core.
What do height limits do? The restrict buildings over X stories. Thus more buildings less than or equal to X stories are built over a greater footprint if demand is fixed. In other words height limits reallocates development. The consequence is that a larger area is urbanized at a higher density (at or near X stories). In DC, there is a much larger urban sphere than, say, height-limit-less Minneapolis, where high-rises in downtown are surrounded by many low-rise and surface parking lots.
Instead of having 10 blocks of 50 story buildings, DC has 50 blocks of 10 story buildings. Is this a really worrisome outcome?
This additional urbanized space is a positive externality in a number of ways. Better urban form (more sidewalks are walkable), less congestion (traffic is spread out over more space), less pollution intake ("the solution to pollution is dilution", the bad stuff is spread over more area), less crime (more eyes near street level), more serendipitous random meetings on the street (which supposedly create greater productivity) and so on.
At one limit, we could have a height limit of 1 story, and spread everything out, at the other, we could have no limit, and buildings would be as concentrated as the market and structural engineering could support. Clearly the first is extreme, but so is the second, so long as we have unpriced externalities. We live in an imperfect (second-best) world with many unpriced externalities (congestion and pollution among them), which have no clear property rights. Regulating heights is one of many second-best solutions to this problem.
Do the height limits imply more suburban development? Sure, someone who really really wants a high rise for some reason will have to locate in the suburbs.
I can't think of a good reason except ego for needing a high rise, while I see many inefficiencies associated with tall buildings: greater distance to the ground floor and thus to people in other buildings (in the absence of skyways on the 50th floor), limited interactions on the upper stories, so much floor space devoted to elevators, higher building costs, etc.)
Otherwise suburban development is not for lack of space in DC, but rather due to a preference for the suburbs. Cities without height limits get their share of suburban development for all the usual reasons (lower land costs, easier access for workers, etc.) when day-to-day inter-firm accessibility is not particularly valuable in their sector, and intra-firm accessibility still matters.